1. Introduction
The cloudtop entrainment instability (CTEI; Deardorff 1980) is considered a major potential mechanism for the transition of the stratocumulus to the trade cumulus over the marine subtropics [cf. Stevens (2005) as an overview]. The basic mechanism of CTEI resides on a possibility that an environmental air entrained into the cloud from the top can be dry enough so that its mixing with the cloudy air leads to evaporation of the cloud water, and induces a sufficient negative buoyancy, leading to further entrainments of the environmental air from the cloud top. The process is expected to finally lead to a transition of stratocumulus into cumuli. A critical review of this process is provided by Mellado (2017), with the review itself even refuting CTEI as further discussed in the end in section 5. Bretherton and Wyant (1997) and Lewellen and Lewellen (2002) propose the decoupling as an alternative theoretical possibility.
However, the existing literature examines CTEI, mostly, in terms of a local condition, such as a buoyancy anomaly at the cloud top (inversion height). Such a parcelbased analysis leads to a criterion for instability in terms of a sign of buoyancy (e.g., Deardorff 1980; Randall 1980; MacVean and Mason 1990; Duynkerke 1993). This type of approaches does not provide a full dynamical picture of the instability, including a quantitative estimate of a growth rate as a function of a horizontal scale (or a wavenumber), and a spatial structure of a preferred instability mode.
The qualitative nature of the existing criteria for CTEI makes it also difficult to test these criteria observationally (cf. Albrecht et al. 1985; Albrecht 1991; Kuo and Schubert 1988; Stevens et al. 2003; Mathieu and Lahellec 2005; Gerber et al. 2005, 2013, 2016). Most fundamentally, a finite time would be required for CTEI to realize. Unfortunately, bulk of existing theories does not tell how long we have to wait to observe CTEI.
A fundamental limitation of existing CTEI studies arises from a fact that these analyses concern only with a sign of a local buoyancy (or vertical eddy buoyancy flux), without properly putting it into a framework of the hydrodynamic instability (cf. Drazin and Reid 1981). Such a dynamically consistent theoretical analysis of the instability couples a given local instability with a full hydrodynamics. It is a standard approach in the midlatitude largescale dynamics to interpret the synoptic cyclones in this manner in terms of the baroclinic instabilities (cf. Hoskins and James 2014). In the author’s knowledge, a hydrodynamic stability analysis is still to be performed for CTEI, probably an exception of Mellado et al. (2009; cf. section 2d). That is the basic approach of the present study.
The hydrodynamics stabilityanalysis method adopted here treats the evolution of the height of the inversion at the top of the mixed layer explicitly with time so that, in principle, its evolution until an ultimate transform into a cumulus regime can be evaluated. For preparing a way for such full analyses, the present study introduces a linear analysis method by taking the dry atmosphere as a demonstrative example. Thus, an important purpose of the study is to show how dynamically consistent instability analyses can be performed in problems of atmospheric boundary layers. The author expects that more studies will follow along this line for better elucidating the dynamics of the cloudtopped boundary layers. Importantly, the study is going to show that even in absence of evaporative cooling, the mixed layer can be destabilized by the entrainment from the top.
It may be considered questionable to perform a linear stability analysis on a fully turbulent system such as the boundary layers. To circumvent this difficulty, the present study assumes that the main role of fully developed convective turbulence is to maintain a vertically wellmixed state of the boundary layer, and that a boundary layer–deep explicit perturbation can be considered as a linear superposition on the mean state maintained by these turbulent flows, however, without explicitly taking into account of the latter.
For this reason, especially, for describing the buoyancy in the wellmixed layer, only an equation averaged over the mixed layer is considered. Note that as a consequence of active vertical mixing, the wellmixed layer is neutrally stratified, in average; thus, no linear buoyancy anomaly can be generated by linear perturbation flows. In this manner, the role of convection remains completely implicit in the present study. However, it is important to keep in mind that the entrainment, that drives the instability, is also driven by convection. Thus, the instability considered herein is ultimately driven by convection.
More specifically, the present study examines a perturbation growth of a mixedlayer deep circulation. This approach is contrasted with some studies, dealing CTEI primarily as a process of generating kinetic energy for smallerscale eddies, that directly contribute to vertical eddy transport at the top of the wellmixed layer associated with entrainment (e.g., Lock and MacVean 1999; Katzwinkel et al. 2012). An overall approach of the present study may be compared with that for the mesoscale entrainment instability by Fiedler (1984; see also Fiedler 1985; Rand and Bretherton 1993). As a major difference, the entrainment induces negative buoyancy by a downward displacement of inversion in the present study, whereas Fiedler considered an enhancement of cloudyair positive buoyancy by entrainment of stable upperlevel air. At a more technical level, the present study considers a change of the buoyancy jump crossing the inversion with time, but fixing the entrainment rate. In Fiedler (1984), in contrast, the main role of the inversion jump is to constrain the entrainment rate.
More general words may be required for some readers who are not familiar with the basics of the hydrodynamic stability analysis. To perform a hydrodynamic stability analysis in a general manner, certain simplifications are always necessary. In this respect, the hydrostatic stability does not pursue any “realism” in the same sense as with both operational and research models widely available today. However, our experience says that those simplified theoretical studies provide useful, and often quantitative information on the process in concern (cf. Pedlosky 1987; Hoskins and James 2014).
The formulation, that couples a standard mixedlayer description with a full hydrodynamics, is introduced in the next section. A perturbation problem is developed in section 3, and some simple solutions are presented in section 4. The paper concludes with the discussion in the last section. An alternative formulation is considered separately in appendix A.
2. Formulation
A dry wellmixed boundary layer is considered. Nevertheless, as we remark from time to time, to some extent, the formulation may also be, at least, conceptually applied to the stratiformtopped mixed layer.
a. Motivations
Essence of CTEI is that a mixing of the freetroposphere air from the above with a cloudy air within stratocumulus leads to evaporation of cloud water due to a dry and relatively high temperature of the entrained freeatmospheric air, but the evaporative cooling, in turn, makes the entrained air colder than the surrounding stratocumuluscloud air, leading to a convective instability that drives the evaporated mixed air farther downward (Deardorff 1980; Randall 1980). Though less frequently considered, a possible reverse process is an intrusion of the cloudy air from the stratocumulus cloud into the free troposphere (e.g., MacVean and Mason 1990; Duynkerke 1993). In this case, when the detrained air is moist enough, it can be more buoyant than the environment due to the virtual effect. Buoyancy induces a further ascent, the ascent leads to adiabatic cooling, the cooling may lead to further condensation of water vapor, and resulting condensative heating can drive the cloudy air farther upward.
Being motivated by investigating this type of instability fully dynamically, first of all, the present study explicitly describes the deformation of the inversion height with time, associated with the entrainment of warm and drier air from the free atmosphere above. The resulting deformation may ultimately lead to transform into a cumulus regime. We will consider the associated processes under a drastically simplified dry mixedlayer formulation. In spite of these drastic simplifications, we somehow recover some basic features of the CTEI just described. The drastic simplification facilitates the analysis of the coupling of these processes with a full dynamics in a form of hydrodynamic stability analysis.
A simple dry mixedlayer formulation is introduced in the next two subsections. It is coupled with a full hydrodynamics introduced in sections 2d and 2e.
b. A mixedlayer formulation for the buoyancy
We consider a wellmixed boundary layer with a depth (inversion height) h. The basic model configuration is shown in Fig. 1. As the most drastic simplification here, we adopt the standard formulation for the dry boundary layer, in which only the buoyancy b vertically averaged over the full mixed layer is considered. This approach is well justified for the dry boundary layer, because the buoyancy phenomenologically is known to be vertically well mixed, as also suggested in Fig. 1.
However, this assumption clearly breaks down for the cloudtopped wellmixed boundary layer. Under standard formulations (e.g., Deardorff 1980; Schubert et al. 1979), the buoyancy anomaly is expressed by a linear relationship with the two conservative quantities, say, the equivalent potential temperature and the total water, which are expected to be vertically well mixed. However, the buoyancy is not expected to be vertically well mixed, because the coefficients for this linear relationship are height dependent [cf. Eq. (3.15) of Schubert et al. (1979), Eqs. (15) and (22) of Deardorff (1976)].
More seriously, although the buoyancy in the dry convective layer may vertically be well mixed in undisturbed state, once a perturbation is applied, a nonvanishing buoyancy disturbance is generated, which we will consider explicitly in the following. Under the latter situation, the buoyancy homogeneity assumption no longer applies. Thus, in more general situations with presence of clouds as well as disturbances, the buoyancy is no longer vertically homogeneous distributed. Nevertheless, as going to be shown in the following, a selfconsistent formulation of the problem is still possible in terms of the vertically averaged buoyancy 〈b〉 even under an explicit presence of the buoyancy perturbations.
Here, we have introduced the variables as follows: t the time, x a single horizontal coordinate considered, u the horizontal wind velocity,
Here, the standard CTEI criteria (Deardorff 1980; Randall 1980) require
c. Perturbation formulation and instability mechanism
Equation (2.9a) contains the two competitive processes arising from the mixedlayertop entrainment: the first is a mechanical mixing as its direct consequence, that leads to a damping, as indicated by the last term on the lefthand side. The second is a consequence of the inversionheight displacement, as seen on the righthand side, which may induce instability. The first effect is independent of scales, whereas the second depends on scales, as further discussed with Eq. (3.8a) below. The scale dependence of the latter leads to a scale dependence of the instability growth as will be shown in section 4.
d. Basic state
To introduce a hydrodynamics, we adopt a twolayer system with constant densities (cf. Fig. 1), closely following a standard formulation for the analysis of the Kelvin–Helmholtz instability as presented, e.g., in chapter 4 of Drazin and Reid (1981). The first layer with a density ρ_{1} represents the wellmixed layer below, and the second with a density ρ_{2} the free troposphere above. To some extent, this formulation can be considered a local description of the dynamics around the top of the wellmixed layer (the inversion height), z = h, although the bottom (surface: z = 0) and the top (z → +∞) boundary conditions are considered explicitly in the following. A height dependence of the density can be introduced to this system, and so long as the densitygradient scale is much larger than a vertical scale of the interest, the given system is still considered a good approximation. Under this generalization, for the most parts in the following, the density values ρ_{1} and ρ_{2} refer to those at the inversion height, z = h. We also assume that the horizontal winds, given by U_{1} and U_{2}, are constant with height in each layer. Thus, we may reset U_{1} = 〈u〉 in the formulation of the last subsection.
Here, an assumed sharp interface is a necessary simplification for treating the essential features of the CTEI in lucid manner, although both recent observational (Lenschow et al. 2000; Katzwinkel et al. 2012) and modeling (Moeng et al. 2005) studies show that the inversion actually constitutes a finitedepth layer with rich morphologies. Mellado et al. (2009) consider a Rayleigh–Taylor instability problem by inserting a positive density anomaly over this thin inversion layer. Their study may be considered an extension to three layers of the present formulation. However, in contrast to the present study, the fluid density is assumed a passive scalar and no possibility of its change associated with the entrainment.
Finally, the basic state
e. Perturbation dynamics

u′→ 0 as$z\to +\phantom{\rule{0.27em}{0ex}}\infty ,$

w′ = 0 at the bottom surface$z=0,$

the pressure is continuous by crossing the inversion, z = h; thus,
3. Stability analysis
The perturbation problem is solved for the dynamics and the buoyancy separately in the following two subsections. Each leads to an eigenvalue problem.
a. Dynamics problem
b. Buoyancy problem
c. Eigenvalue problems
As the analysis of the last two subsections show, the stability problem reduces to that of solving the two eigenvalue problems given by Eqs. (3.7) and (3.9). Here, the problem consists of defining two eigenvalues: the growth rate σ and the vertical wavenumber m of the mixed layer for a given horizontal wavenumber k. Thus, two eigenequations must be solved for these two eigenvalues.
In the following, we first nondimensionalize these two eigenequations, then after general discussions, derive a general solution for the growth rate obtained from a nondimensionalized version of Eq. (3.7). This solution has a general validity. It also constitutes a selfcontained solution when a coupling of the dynamical system considered in sections 2c and 3a with the buoyancy is turned off by setting α = 0 in Eq. (2.9a).
A convenient general strategy for solving this set of eigenequations would be to first solve Eq. (3.12a) for
4. Simple solutions
a. Simplest case
The remainder of this subsection provides a selfcontained mathematical description of how a closed analytic solution is derived. Readers who wish only to see the final results may proceed directly to the last two paragraphs of this subsection.
Note that
Nondimensional growth rate
Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JASD210246.1
Nondimensional growth rate
Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JASD210246.1
Nondimensional growth rate
Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JASD210246.1
Recall that this solution is derived under an approximation of Eq. (4.2). Under this approximation, we seek a solution with convective plumes in the mixed layer slightly intruding into the free troposphere (cf. Fig. 3), as inferred by examining the assumed solution forms (3.2a)–(3.2d). By combining this fact with the phase relations between the variables already identified Eqs. (3.3a), (3.3b), (3.4a), (3.4b), (3.5a), (3.5b), (3.6a), and (3.6b), we can easily add spatial distributions of the other variables to Fig. 3, as already outlined after Eq. (4.1a) in section 4a.
Schematic structure of the perturbation solution: the streamfunction ψ (contours) and the inversionheight deformation (thick solid curve).
Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JASD210246.1
Schematic structure of the perturbation solution: the streamfunction ψ (contours) and the inversionheight deformation (thick solid curve).
Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JASD210246.1
Schematic structure of the perturbation solution: the streamfunction ψ (contours) and the inversionheight deformation (thick solid curve).
Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JASD210246.1
b. Largescale divergence effect
The simplest case considered in the last subsection illustrates how an instability can be induced by the mixedlayertop entrainment. However, the setting is rather unrealistic by neglecting a contribution of the largescale divergence rate
c. Under steady deepening by entrainment
5. Discussion
A hydrodynamic stability analysis has been applied to the dry convective wellmixed boundary layer with an ultimate application of the methodology to the CTEI in mind. The key difference of this approach from the more conventional parcelbased analysis is that it can derive the growth rate of an instability as a function of the horizontal wavenumber as well as its spatial structure.
The analysis has identified a new type of instability associated with the mixedlayertop entrainment. This instability arises under a competition between the destabilization tendency due to the entrainmentinduced buoyancy perturbation and the stabilization tendency due to the mechanical damping associated with entrainment. Importantly, these two entrainment effects can be separated into these the two terms in the buoyancy equation, Eq. (2.9a). Damping tendency of the buoyancy perturbation is directly proportional to the entrainment velocity w_{e}, whereas the destabilization tendency by buoyancy perturbation is generated by the displacement of the interface, which is more directly controlled by another parameter α [Eq. (2.9b)]. This instability is, more fundamentally, driven by the entrainment, which is in turn, driven by convection in the wellmixed layer.
Obtained growthrate tendencies with changing horizontal scales are consistent with qualitative arguments in section 3 associated with Eq. (3.10). In the smallscale limit, the mechanical damping effect dominates over the destabilization tendency by buoyancy perturbation, and as a result, the perturbation is always damping. In the largescale limit, instability may arise when the magnitude of the destabilization tendency by buoyancy feedback is stronger than the mechanical damping as measured by a ratio between the two parameters
The identified instability is inherently of a largescale nature, and a reasonably large domain is required to numerically realize it, as suggested by Fig. 2. If this instability had any implication for the CTEI, it could explain why the evidence for the CTEI by LES studies so far is rather inconclusive (e.g., Kuo and Schubert 1988; Siems et al. 1990; MacVean 1993; Yamaguchi and Randall 2008). In these simulations, relatively small domain sizes (5 km square or less) are taken, that may prevent us from observing a full growth of the CTEI. The obtained growth time scale is also very slow, about an order of a day. With typically short simulation times with LESs (about few hours), that could be another reason for a difficulty for realizing a CTEI with these simulations. Direct numerical simulations (DNSs) by Mellado (2010), in spite of an advantage of resolving everything explicitly, are even in less favorable position for simulating a full CTEI due to an even smaller modeling domain. Unfortunately, dismissal of a possibility of CTEI by Mellado (2017) in his review is mostly based on this DNS result.
In contrast to these more recent studies, it may be worthwhile to note that an earlier study by Moeng and Arakawa (1980) identifies a reasonably clear evidence for CTEI over a high sea surface temperature (SST) region of their twodimensional nonhydrostatic experiment with a 1000km horizontal domain, assuming a linear SST distribution. A preferred scale identified by their experiment is 30–50 km, qualitatively consistent with the present linear stability analysis, although it is also close to the minimum resolved scale in their experiment due to a crude resolution. A time scale estimated from the present study is also consistent with a finding by Moeng and Arakawa (1980) that their CTEIlike structure develops taking over 24 h. However, due to limitations of their simulations with parameterizations of eddy effects, a full LES is still required to verify their result. From an observational point of view, an assumption of horizontal homogeneity of the stratocumulus over such a great distance may simply be considered unrealistic in respect of extensive spatial inhomogeneity associated with the stratocumulus as realized in LESs (e.g., Chung et al. 2012; Zhou and Bretherton 2019).
In this respect, it may be interesting to note that a recent observational study by Zhou et al. (2015) suggests a possibility of a certain cloudtop instability, if not CTEI, leading to a decoupling, which ultimately induces a transition to trade cumulus regime. We should realize that a slow time scale suggested for CTEI by the present study may be another reason for difficulties of identifying it observationally. Previous observational diagnoses on CTEI criterions have been based on instantaneous comparisons (e.g., Albrecht et al. 1985; Albrecht 1991; Kuo and Schubert 1988; Stevens et al. 2003; Mathieu and Lahellec 2005; Gerber et al. 2005, 2013, 2016). A finite time lag could be a key missing element for a successful observational identification of CTEI. If that is the case, data analyses from a point of view of the dynamical system as advocated by Yano and Plant (2012) as well as Yano et al. (2020) becomes a vital alternative approach.
On the other hand, although the present analysis has been performed by assuming a dry atmosphere, it is less likely than in the marine stratiformtopped boundary layers that this instability is to be seen in dry wellmixed convective boundary layers. The latter typically go through very pronounced diurnal cycles with the boundary layer itself becomes stably stratified during nights; thus, a good stationarity of the system required to observe such a slow growth of instability is hardly satisfied.
The present study focuses on the instability induced by the boundary layer–top entrainment. Nevertheless, a basic formulation is presented in fully general manner. Thus, its simple extension can consider rich possibilities of the mixedlayer inversioninterface instabilities under a coupling with the buoyancy anomaly. Especially, the present formulation allows us to explicitly examine a possibility of the Kelvin–Helmholtz instability over the mixedlayer observationally suggested by Brost et al. (1982), Kurowski et al. (2009), Katzwinkel et al. (2012), and Malinowski et al. (2013).
Furthermore, the present analysis of the dry convective wellmixed layer constitutes a first step to fully examine the CTEI as a hydrodynamicinstability problem, most importantly, by explicitly introducing the evaporative cooling effect (cf. de Lozar and Mellado 2015). Other types of possible instabilities in the cloudtopped boundary layers, such as decoupling (Bretherton and Wyant 1997; Lewellen and Lewellen 2002), can equally be addressed by the present framework.
Extensive physics can also be incorporated. In this respect, LESs by Yamaguchi and Randall (2008) can be instructive: although their idealized version of LES leads to a positive feedback suggesting CTEI, the tendency is overcompensated by longwave radiation and surface heat flux in simulations with full physics. LES studies also show that the cloudtop entrainment rate is sensitively modified under aerosol–cloud (Xue et al. 2008; Hill et al. 2008, 2009) and cloud–radiation interactions (Zhou and Bretherton 2019). The present formulation provides a basis for elucidating those various feedbacks between physics under the framework of the linearstability analysis.
A crucial aspect of the present formulation is to treat a deformation process of the inversion interface explicitly, that could ultimately transform the wellmixed layer into a cumulus regime. The main original contribution of the present study is, under a crude representation of the wellmixed layer, to present its linear growth rate as a function of the horizontal scale. More elaborated studies would certainly be anticipated, and the present study suggests that they are actually feasible. A more elaborated entrainment formulation (cf. Stevens 2002) is just one example. The most challenging step is to proceed to a fully nonlinear formulation, probably, by taking an analogy with the contour dynamics for the vortex dynamics (cf. Dritschel 1989; Dritschel and Ambaum 1997), but by considering a full nonlinear evolution of the inversion height as a contour. Such an extension would be able to simulate a transformation of stratocumulus into trade cumulus in terms of a finite amplitude deformation of the inversion height. Both modeling and observational studies are further expected to follow.
Acknowledgments.
Chris Bretherton led my attention to Fiedler (1984), Bjorn Stevens to Mellado (2017), and Szymon Malinowski to Zhou et al. (2015). I would also like to thank the editor in charge, David Mechem, for his enduring efforts leading to publication of the present manuscript. Many reviewers have also contributed to this process, but I would like to specifically thank an anonymous reviewer who followed the series of revisions of the manuscript to the end. I specifically refer to this reviewer at several places of the text for this reason.
Data availability statement.
The present study does not use any data either generated numerically nor by observation nor by laboratory experiments. Programs used for generating graphics are available by request.
APPENDIX A
Alternative Formulation with Entrainment Perturbation
Here, a positive perturbation buoyancy,
Asymptotic expansions as in the main text have also been attempted, but without success. Thus, eigensolutions are numerically sought over the range of
The result is summarized in Fig. A1. Here, only the most prominent mode (i.e., fastest growing or least damping) is plotted both for diverging and converging background flows. In the divergentflow regime (i.e.,
Nondimensional growth rate
Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JASD210246.1
Nondimensional growth rate
Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JASD210246.1
Nondimensional growth rate
Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JASD210246.1
Note that in the convergingflow regime (i.e.,
APPENDIX B
Typical Physical Values
Typical physical values (in the orders of magnitudes) of the problem are as follows:

Acceleration of the gravity: g ∼ 10 m s^{−2}

Entrainment rate: w_{e} ∼ 10^{−2} m s^{−1} (cf. Stevens et al. 2003; Gerber et al. 2013)

Inversion height:
$\overline{h}\sim {10}^{3}\hspace{0.17em}\text{m}$ (cf. Schubert et al. 1979)
We also set
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